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# Suppose is a solution to . What is the sum of and its reciprocal?

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Suppose $$x$$ is a solution to $$x^2 + 1 = 7x$$ . What is the sum of $$x$$ and its reciprocal?

Oct 22, 2019

#1
+1078
0

You can simply find x and calculate

Eventually, this equals 7.

However, there is a faster way. Can you find out yourself?

You are very welcome!

:P

Oct 22, 2019
#2
0

I think using the quadratic formula, we can find what x is equal to.

so

x^2+1=7x

Subtract 7x from both sides

x^2-7x+1

(Notice it is in the form ax^2+bx+c)

$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$

You will eventually get

x=6.85410196 approx.=6.85

Now I don't understand what do you mean what is the sum of x?

Its reciprocal is just 1/6.85 i guess.

Oct 23, 2019
#3
+109563
+1

Rearrange as

x^2  - 7x  = -1

Complete the square on x

x^2  - 7x  +  49/4  =  -1 + 49/4

(x - 7/2)^2  =  45/4       take the positive root

x - 7/2  =   √45 / 2

x - 7/2   =  3√5/2

x  =  [ 7 + 3√5]  / 2

Its reciprocal is    2  / [ 7 + 3√5]

So

[ 7 + 3√5]  / 2  +   2  / [ 7 + 3√5]   =

[ 7 + 3√5] ^2  + 4

_______________   =

2  [ 7 + 3√5]

[ 49  + 42√5 + 45 + 4 ]

__________________   =

2  [ 7 + 3√5]

[  98 + 42 √5 ]

____________  =

2  [ 7 + 3√5]

49 + 21√5

_________  =

7 +  3√5

[49 + 21√5] [ 7 - 3√5]

__________________   =

49  - 45

343  + 147√5 - 147√5 - 63*5

_______________________  =

4

343  -  315

_________   =

4

28

__  =

4

7

Oct 23, 2019